What is the distance in meters?

So far, the closest anyone has come to a mathematical derivation (ab initio) of the distance between two atoms in a molecule is for the simplest case of them all. This is NOT the hydrogen molecule itself, which is already too complicated for that (it includes two electrons) but the "molecular ion" H2+ which consists of two protons and a SINGLE electron binding them. See first link below for the presentation by Mark E. Tuckerman (professor of chemistry at NYU).

Tuckerman's discussion is based on nonrelativistic quantum mechanics (see second link for a general introduction by myself).

The distance you are after (for H2+ at least) is denoted R_e and called "equilibrium bond length" near the bottom of Tuckerman's page. He does not give a numerical value which could be compared to experiment.

That comparison is done at the bottom of another page (third link below) by Richard Fitzpatrick (professor of physics at UT Austin):

For H2+, the theoretical prediction is 1.33e-10 m

and the experimental value is 1.06e-10 m.

For the neutral hydrogen molecule itself, all we have are fairly accurate experimental values (see next-to-last link below, which follows a general discussion of fundamental constants).

Finally, energetic amateurs have been trying to extend Bohr's semi-classical model to deal with multiple electrons. The "law of small numbers" applied to such formulas may bless them with some success where no success is deserved (i.e, there are only so many possible approximative formulas, so one of them can turn out to be fairly good even if based on the wrong concepts). Arguably, Bohr's own "old quantum theory" owed some of its early success to that kind of luck too, at least in part...

I just stumbled on one such effort as I was researching this post. I am linking to that obscure site although I did not spend much time reviewing it (I do feel quite uncomfortable with some of the author's remarks). See the last link below as an entry point to a set of several pages which may or may not prove dubious in the end...

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